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2 ORDER PROPERTIES OF THE REAL NUMBERS 19
(2.2.5) If(x^) and (yt) are two finite sequences ofn real numbers (1 < / < n)
such that xt ^ ytfor each /, then |
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7/7/7 addition xt < ytfor one index i at least, then
x1 + x2 + - - • +xn < yl + y2 + - • • + yn .
For n = 2 the assumptions imply successively by (II.4)
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hence the first conclusion in that case; moreover, the relation xl-\-x2
= y\ + J2 implies x^ 4- x2 = ^ + j2 = yt + >>2 , hence *2 = j;2 and Xi=yl9 from which our second statement follows. The proof is concluded by induc- tion on /7, applying the result just obtained for n = 2. |
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(2.2.6) The relation x ^ y is equivalent to x -f z < 7 + z; same result when
< is replaced by <.
We already know by (IIA) that x < y implies x + z < y + z; conversely
^ 4- z < y -f z implies x + z + ( — z) ^ y + z 4- (—z), i.e. x < j. On the other hand, x + z = j-fzis equivalent to x = y. |
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(2.2.7) The relations x^y9 Q ^ y — x, x — j^^O, — y ^ — x are equivalent;
same result with < replacing <.
This follows from (2.2.6) by taking in succession z = — x, z = — y
and z = — x — y.
Real numbers such that x^Q (resp. x > 0) are called positive (resp.
strictly positive); those which are such that x^O (resp. x < 0) are called negative (resp. strictly negative). The set of positive (resp. strictly positive) numbers is written R+ (resp. R+). |
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(2.2.8) If Xi,...,xn are positive, so is x1 + x2 + •• • 4- xn; moreover
xi + X2 + "" + xn > 0 unless xl = x2 = * * • = xn = 0.
This is a special case of (2.2.5).
In particular, x ^ 0 (resp. ;c> 0) is equivalent to 77 • x ^ 0 (resp. n • x > 0)
for any integer n > 0. |
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